Economic Models - Convex Optimization

30 Alexis Lazaridis
5.1. Testing for Co-integration
Since for the first two cases, ∑ û i ̸= 0, we estimated the following regression:
q∑
û i = b 1 + b 2 û i−1 + b j+2 û i−j + ε i . (17)
The value of q was set such that the noises ε i to be white, as it will be
explained in detail later on. Regarding the last case, the intercept is omitted
from Eq. (17), since û i ’s are OLS residuals, so that ∑ û i = 0.
The estimation results are as follows:
u ˆml i =−0.00612 − 0.381773 uml i−1 − 0.281651uml i−1
(0.193688)
j=1
t =−3.682
uŝvd i = 0.006525 − 0.428278 usvd i−1 − 0.259066usvd i−1
(0.108269)
t =−3.956
ubôok i =−0.606759 ubook i−1
(0.09522)
t =−6.37.
Since these error series are the results of specific calculations and in
the simplest case are the OLS residuals, it is not advisable (Harris, 1995,
pp. 54–55). to apply the Dickey-Fuller (DF/ADF) test, as we do with any
variable in the initial data set. We have to compute the t u statistic from McKinnon
(1991, p. 267–276) critical values (see also Harris, 1995, Table A6,
p. 158). This statistic (t u = ∞ + 1 /T + 2 /T 2 ) for three (independent)
variables in the long-run relationship with intercept is:
α = 0.01 α = 0.05 α = 0.10
t u =−4.45 t u =−3.83 t u =−3.52.
Hence, {book i } is stationary for α = (0.01, 0.05, 0.10), {usvd i } is stationary
for α = (0.05, 0.10) and {uml i } is stationary only for α = 0.10.
Recall that the null [û i ∼ I(1)] is rejected in favor of H 1 [û i ∼ I(0)], if
t